DE112017002262T5 - Designing FIR filters with global minimax optimal magnitude response - Google Patents

Abstract

Embodiments of the present disclosure provide mechanisms that enable a design of an FIR filter having a guaranteed global optimal magnitude response with respect to the minimax optimality criterion, with the desired weighting on the error in the stopband versus the passband. The design of such a filter is based on a theorem ("characterization theorem") which provides an approach for characterizing the global minimax optimality of a given FIR filter h [n], n = 0, 1, ..., N, because the Optimality with respect to an magnitude response of this filter, | H (e) |, as compared to the desired filter response, D (ω), which is one in the passband and zero in the stopband. The characterization theorem allows characterizing an optimality for both real and complex valued filter coefficients and does not require any symmetry in the coefficients, thereby being applicable to all non-linear phase FIR filters.

Description

Cross-reference to related applications

This application claims the benefit and priority of US Provisional Patent Application Serial No. 62 / 330,084, filed April 30, 2016, and entitled "DESIGNALING FIR FILTERS WITH GLOBALLY MINIMAX-OPTIMAL MAGNITUDE RESPONSE", and US Serial Number Serial Number Application No. 15 / 271,487 filed Sep. 21, 2016, entitled "DESIGNAL FIR FILTERS WITH GLOBALLY MINIMAX-OPTIMAL MAGNITUDE RESPONSE", which is hereby incorporated by reference in its entirety.

Technical field of the disclosure

The present invention relates to the field of digital signal processing, in particular the design of Finite Impulse Response (FIR) filters.

background

As is well known, an electromagnetic receiver is an electronic device which receives electromagnetic waves in a certain frequency range and converts the information carried by these waves into any kind of usable form. For example, a receiver, typically referred to as a "radio receiver," receives electromagnetic waves in the radio range from approximately 3 kilohertz (kHz) to 300 gigahertz (GHz). All receivers use antennas to detect the waves and convert them into alternating current (AC) signals and electronic filters to separate the signals in the desired frequency band from all other signals that can be detected by the antenna. In the context of receivers, different frequency bands are sometimes referred to as "channels".

Selectivity performance of a receiver refers to a measure of the ability of the receiver to separate the desired frequency range (termed a "passband Ω _P " of frequencies ω) from unwanted spurious signals occurring at frequencies other than a "stopband Ω _S " of frequencies ω to be received). In other words, selectivity defines how effectively a receiver can only respond to the signal of interest it is tuned to (ie the signal in the desired frequency band) and can suppress signals in other frequencies.

Filters can be classified into different groups depending on which classification criteria are used. Two major types of digital filters are FIR (Finite Impulse Response) and IIR (Infinite Impulse Response) filters, each with its own advantages and disadvantages.

An FIR filter is designed by finding coefficients and filter orders that can meet certain specifications. In other words, in a filter design environment, "filter design" refers to determining a filter order N and determining values of (N + 1) coefficients h [n] of a filter that would approximate the ideal response given by the constraints in both the passband as well as defined in the stopband. In this context, a filter order N is a positive integer, and for each coefficient, n is an integer of a sequence of consecutive integers from 0 to N (i.e., n = 0, 1, ..., N). Accordingly, for a second-order filter (i.e., N = 2), coefficients may be referred to as h [0] and h [2].

The specifications of an ideal response that is to satisfy a filter being designed are typically expressed based on the desired selectivity performance of a receiver. Such constraints could be expressed in terms of a frequency response H (e ^jω ) (ie, a Fourier transform of the impulse response h [n]) provided with the passband Ω _P and the stopband Ω _S of frequencies to obtain the desired magnitude response D (ω ) to approximate: $$\text{D}\left(\mathrm{\omega }\right)=\{\begin{array}{c}\mathrm{1,}\mathrm{\omega }\in {\mathrm{\Omega }}_p\\ 0\mathrm{\omega }\in {\mathrm{\Omega }}_s\end{array}$$

wobei K_des ein positiver Skalar ist, der als ein Teil der Filtervorgaben gegeben ist. Bereitstellen einer Gewichtung größer als Eins in dem Sperrband platziert einen Schwerpunkt darauf, eine bessere Approximation für die ideale Antwort in dem Sperrband zu haben (d. h., das gestaltete Filter sollte die Frequenzen des Sperrbandes adäquat unterdrücken).Another constraint could include a desired weighting function W _of (ω) (where the index "des" is a "desired" abbreviation) that dictates the relative centroid of the error in the stopband as compared to the passband. In particular, the weighting requirement could be expressed as $${\text{W}}_{des}\left(\mathrm{\omega }\right)=\{\begin{array}{ll}\mathrm{1,}\hfill & \mathrm{\omega }\in {\mathrm{\Omega }}_p\hfill \\ K_{des}.\hfill & \mathrm{\omega }\in {\mathrm{\Omega }}_s\hfill \end{array}$$

wherein _the K is a positive scalar, which is given as a part of the filter specifications. Providing a weight greater than one in the stopband places a focus on having a better approximation for the ideal response in the stopband (ie, the designed filter should adequately suppress the stopband frequencies).

Many FIR filter design methods exist, such as. A windowing method, frequency sampling method, weighted least squares design, a Parks-McClellan method, etc., all of which attempt to arrive at the filter coefficients h [n] of a filter providing an ideal filter response provided by the constraints, best approximated. Some of these methods can guarantee that for a given value of the filter order N and certain conditions imposed for h [n], the result is the best possible approximation for these conditions. For example, applying a rectangular window of size N + 1 to the ideal filter response results in the best approximation in terms of the root mean square error optimality criterion. Another example is that if the filter is constrained to be symmetric about its center index, the Parks-McClellan filter design method yields the best minimax approximation, ie, minimizes the maximum error. However, finding coefficients for FIR filters with nonlinear phase characteristics, ie, for the most general form of FIR filters, where the phase response of a filter may be a nonlinear function of frequency, remains a challenge if the optimality criterion is to minimize the maximum error , Improvements could be made with respect to addressing one or more of these problems.

list of figures

• 1A veranschaulicht ein Beispiel für eine Betragsantwort (|H(e^jω)|, d. h. des Absolutwerts) der Frequenzantwort (Fourier-Transformation) des Filters h[n] als eine Funktion von ω gemäß manchen Ausführungsformen der Offenbarung;
• 1B veranschaulicht ein Beispiel für eine Frequenzantwort (P(e^jω))) einer Folge p[n] als eine Funktion von ω gemäß manchen Ausführungsformen der Offenbarung;
• 2 veranschaulicht ein Beispiel für eine Frequenzantwort (G(e^jω))) einer Folge g[n] als eine Funktion von ω gemäß manchen Ausführungsformen der Offenbarung;
• 3 stellt ein Flussdiagramm eines Verfahrens zum Berechnen von Koeffizienten h[n] eines FIR-Filters der Ordnung N mit einer Minimax-optimalen Betragsantwort |H(e^jω)| gemäß manchen Ausführungsformen der Offenbarung bereit;
• 4 stellt eine schematische Veranschaulichung eines Empfängers gemäß manchen Ausführungsformen der Offenbarung bereit; und
• 5 stellt ein Blockdiagramm dar, das ein beispielhaftes Datenverarbeitungssystem gemäß einer Ausführungsform der vorliegenden Offenbarung veranschaulicht.

In order to provide a more complete understanding of the present disclosure and the features and advantages thereof, reference is now made to the following description, taken in conjunction with the accompanying drawings, wherein like reference numerals represent like parts, in which:

• 1A FIG. 12 illustrates an example of an magnitude response (| H (e ^jω ) |, ie, the absolute value) of the frequency response (Fourier transform) of the filter h [n] as a function of ω according to some embodiments of the disclosure;
• 1B illustrates an example of a frequency response (P (e ^jω ))) of a sequence p [n] as a function of ω, in accordance with some embodiments of the disclosure;
• 2 illustrates an example of a frequency response (G (e ^jω ))) of a sequence g [n] as a function of ω according to some embodiments of the disclosure;
• 3 FIG. 10 illustrates a flowchart of a method for calculating coefficients h [n] of an order N-FIR filter with a minimax optimal magnitude response | H (e ^jω ) | according to some embodiments of the disclosure;
• 4 provides a schematic illustration of a receiver according to some embodiments of the disclosure; and
• 5 FIG. 12 is a block diagram illustrating an exemplary data processing system according to an embodiment of the present disclosure. FIG.

Description of embodiments of the disclosure

Minimax optimality challenges for generic FIR filters

A minimax optimality criterion is to minimize the maximum value of the error. In a filter design environment (ie, when computing coefficients of a filter), a designed filter will typically approximate the ideal response in both the passband and the stopband to some extent. Furthermore, it is possible to apply some weight to the stopband so that a larger center of gravity is placed on it to have a better approximation in the stopband. The maximum value of the absolute value of the weighted error in the entire frequency range, including the stopband and the passband, is expressed as | _∞ -norm of the weighted error (| _∞ is pronounced as "EI-infinity"). A minimax filter design refers to the process of finding filter coefficients of a certain order involving the | _∞ errors, ie, will minimize the maximum weighted error encountered in the entire frequency range.

To further explain how to evaluate a minimax-optimal filter, a hypothetical filter, A, can be considered. For the hypothetical filter A, the weighted error can oscillate anywhere between -0.01 and 0.01 on the frequency axis, except that at the frequency ω = 0.30π, the value of the weighted error is 0.20. The | _∞ -norm of this error function is 0.20. Another hypothetical filter, B, can also be considered. For the hypothetical filter B, the error can oscillate anywhere between -0.19 and 0.19, but never becomes larger than that _∞ -norm of this error is 0.19. Although the filter A approximates the ideal response much better than the filter B at almost any frequency, the minimax error criterion favors the filter B over the filter A because the filter B for the worst case scenario (which would occur in this illustrative illustration, if total input at ω = 0.30π would be concentrated) is "safer". Accordingly, a minimax Design considered to be prepared for the worst case scenario.

Linear phase filters (ie, filters for which a phase characteristic is a linear function of frequency) have certain symmetries about their midpoint, which allows their frequency response to be written as a real-valued, zero-phase response multiplied by a linear phase term. Since the alternation theorem and the Remez replacement algorithm only work with real-valued functions, they can be directly applied to characterize or design globally minimax-optimal linear phase filters merely by taking the real part of the frequency response into account. A generic FIR without such symmetry limits would allow better flexibility in selecting its coefficients and may be more advantageous than linear phase filters. However, since the frequency response of a non-linear FIR filter can not necessarily be expressed as a real-valued function multiplied by a linear phase term, as is done for linear phase filters, the Alternation Theorem and Remez exchange methods can not be used for global minimax-optimal design of generic filters. which may include nonlinear filters. There have been attempts in the literature to carry the design problem into the autocorrelation domain of the filter rather than the domain of the filter itself, an idea also used in the present disclosure. However, these approaches in the literature do not follow any well-defined optimality criterion in the filter domain. Instead, these approaches find an optimal filter in the autocorrelation domain which, if converted back to the filter domain, may not optimally remain or may not show the desired focus on the attenuation in the stopband over the pass band given by the weighting function W _of FIG is.

Overview

Embodiments of the present disclosure provide mechanisms that enable implementation of a digital filter that may improve one or more of the problems described above, particularly with respect to designing an FIR filter that would have a guaranteed global optimal magnitude response with respect to the minimax optimality criterion , Designing and then applying such a filter to filter input signals provides a beneficial technological solution to a problem of suboptimal conventional filters (i.e., a technology rooted problem).

The design of such a filter is based on a practical application of a theorem derived by the inventor of the present disclosure. The theorem may be referred to as a "characterization theorem" for representing the fact that it provides an approach for characterizing the global minimax optimality of a given FIR filter h [n], n = 0, 1, ..., N the optimality with respect to the magnitude response of this filter, | H (e ^jω ) |, is judged as compared to the desired filter response, D (ω), which is one in the passband Ω _P and zero in the stopband Ω _S. In particular, the characterization theorem allows judging whether a given filter has a magnitude response that is the best approximation for D (ω) in that no other magnitude response for the same order of the FIR filter can be achieved, which has a smaller infinity norm on the weighted one Error function W _of (ω) (| H (e ^jω ) | - D (ω)). The characterization theorem allows characterizing an optimality for both real and complex valued filter coefficients and does not require any symmetry in the coefficients, thereby being applicable to all non-linear phase FIR filters.

In turn, observations of the characterization theorem allow for an efficient method as described herein for designing nonlinear phase FIR filters in cases where only the magnitude response is predetermined and the phase is not limited. Such a method is referred to herein as a "FIR filter design method". While the FIR filter design method is not limited to a particular phase response, it nevertheless advantageously allows for a desired phase design, e.g. Minimal phase design, without compromising global optimality of the filter with respect to the amount. The next section sets forth the characterization theorem provided for completeness with respect to the mathematical proof of the optimality of the FIR filter design method proposed herein. Thereafter, the filter designing method will be described.

As one of ordinary skill in the art appreciates, aspects of the present disclosure, and in particular the FIR filter design approach described herein, may be implemented in various ways - e.g. As a method, system, computer program product or computer readable storage medium. Accordingly, aspects of the present disclosure may take the form of a fully hardware-based embodiment, a fully software-based embodiment (including firmware, resident software, microcode, etc.) or an embodiment that combines software and hardware aspects, all of which are generally referred to herein as a "circuit,""Module" or "system" can be called. At least Some functions described in this disclosure may be implemented as an algorithm that may be implemented by one or more processing units, e.g. As one or more microprocessors, one or more computers can be executed. In various embodiments, different steps and parts of the steps of each of the methods described herein may be performed by different processing units. Further, aspects of the present disclosure may take the form of a computer program product in one or more computer-readable media, preferably non-volatile, with computer-readable program code embodied thereon, e.g. B. stored. In various embodiments, such a computer program may be downloaded (updated) to existing devices and systems (eg, the existing filter modules, electromagnetic receivers, or controllers of such filters or receivers, etc.) or stored thereon in manufacturing these devices and systems.

Other features and advantages of the disclosure will be apparent from the following description and from the claims and examples.

Charakterisierungstheorem

It is assumed that a filter with coefficients h [n], n = 0, 1,..., N and a frequency response H (e ^jω ) with the passband Ω _P and the stopband Ω _S of frequencies is provided for to approximate the desired magnitude response D (ω) defined by the equation (1) above.

Further, a desired weighting function W _of (ω) is assumed, which specifies the relative center of gravity on the suppression of an error in the stopband as compared to the passband by equation (2) above. The scalar K _des may be given as part of a design specification (that is, provided as an input to the FIR filter design method / algorithm described herein).

$${\delta }_p={\text{max}}_{\omega \in \left({\mathrm{\Omega }}_p\cup {\mathrm{\Omega }}_s\right)}\left|E_w\left(\omega \right)\right|$$

bzw. $${\delta }_S=\frac{{\delta }_P}{K_{des}}$$

The weighted error function E _W (ω) achieved by the designed filter h [n] and the limits of the passband error δ _P (ie, the maximum (absolute) deviation of | H (e ^iω ) | of one in the passband ) and the guard band error δ _S (ie the maximum (absolute) deviation of | H (e ^iω ) | of unity in the stop band) can be defined as $$e_W\left(\mathrm{\omega }\right)={\text{W}}_{des}\left(\mathrm{\omega }\right)\left(\left|H\left(e^{j\omega }\right)\right|-D\left(\omega \right)\right).$$

$${\delta }_p={\text{Max}}_{\omega \in \left({\mathrm{\Omega }}_p\cup {\mathrm{\Omega }}_s\right)}\left|e_w\left(\omega \right)\right|$$

respectively. $${\delta }_S=\frac{{\delta }_P}{K_{des}}$$

wenigstens N+2 Alternationen aufzeigt, falls die Filterkoeffizienten h[n] darauf begrenzt sind, realwertig zu sein, oder 2N+2 Alternationen aufzeigt, falls die Filterkoeffizienten h[n] nicht darauf begrenzt sind, realwertig zu sein; wobei die angepasste gewünschte Gewichtungsfunktion, W’_des (ω), definiert ist als $$W{\text{'}}_{des}\left(\mathrm{\omega }\right)=\{\begin{array}{ll}\mathrm{1,}\hfill & \mathrm{\omega }\in {\mathrm{\Omega }}_p\hfill \\ 2K_{des},\hfill & \mathrm{\omega }\in {\mathrm{\Omega }}_s\hfill \end{array},$$

und die angepasste gewünschte/anvisierte Betragsantwort D’ (ω) definiert ist als $$D\text{'}\left(\mathrm{\omega }\right)=\{\begin{array}{ll}\mathrm{1,}\hfill & \mathrm{\omega }\in {\mathrm{\Omega }}_p\hfill \\ \frac{{\delta }_S}{2},\hfill & \mathrm{\omega }\in {\mathrm{\Omega }}_s\hfill \end{array}.$$

The characterization theorem can then be formulated as follows: | H (e ^jω ) | is the unique minimax optimal magnitude response which can then be obtained by any N order FIR filter to approximate the ideal filter magnitude response D (ω) to a desired weighting function W _of (ω) if and only if the adjusted weighted error function $$e{\text{'}}_W\left(\mathrm{\omega }\right)=W{\text{'}}_{des}\left(\mathrm{\omega }\right)\left(\left|H\left(e^{j\omega }\right)\right|-D\text{'}\left(\omega \right)\right)$$

show at least N + 2 alternations if the filter coefficients h [n] are limited to being real or 2N + 2 alternations if the filter coefficients h [n] are not limited to being real; wherein the adjusted desired weighting function, W ' _of (ω), is defined as $$W{\text{'}}_{des}\left(\mathrm{\omega }\right)=\{\begin{array}{ll}\mathrm{1,}\hfill & \mathrm{\omega }\in {\mathrm{\Omega }}_p\hfill \\ 2K_{des}.\hfill & \mathrm{\omega }\in {\mathrm{\Omega }}_s\hfill \end{array}.$$

and the adjusted desired / targeted magnitude response D '(ω) is defined as $$D\text{'}\left(\mathrm{\omega }\right)=\{\begin{array}{ll}\mathrm{1,}\hfill & \mathrm{\omega }\in {\mathrm{\Omega }}_p\hfill \\ \frac{{\delta }_S}{2}.\hfill & \mathrm{\omega }\in {\mathrm{\Omega }}_s\hfill \end{array},$$

As is known in the art, "alternations" are defined as the frequency points at which the weighted error function reaches its extreme values, with one extreme considered as an alternation if its sign is opposite to the previous extreme and its magnitude is equal to the previous one Extreme value (a positive alternation is followed by a negative alternation with the same amount and vice versa). Generally, an expression "extremum" refers to a local minimum or maximum local value, ie, a point of lower or greater value than its neighbor points. The optimality in linear phase filters is characterized by counting the alternations with respect to the weighted error function calculated using the desired response and the desired weighting function provided. The Characterization provided here for the magnitude response operates in contrast to the frequency response with a matched weighted error function, as described in equation (6) above. Such an adaptation can be proved mathematically, but already intuitively such adaptation seems to be needed because of amounts that can never go below zero. The adjusted desired / target magnitude response D '(ω) in the passband may be chosen as the midpoint of the error band [1- δ _P , 1 + δ _P ] while the adjusted desired / target magnitude response D' (w) in the stopband as the midpoint of the error band [0, δ _S ] can be selected.

The number of alternations needed for complex valued coefficient filters is greater than that of real valued coefficient filters. This is also consistent with intuition because it reflects the additional degrees of freedom in choosing the filter coefficients by relaxing the constraint on being real and can also be proved mathematically.

Proposed FIR filter design

The following describes design algorithms for filters that are limited to having real-valued coefficients. Therefore (N + 2) alternations are needed. However, the justification provided below for complex valued coefficient filters is simple in that (2N + 2) alternations are needed; H. by replacing all occurrences of (N + 2) with (2N + 2).

Observations of the characterization theorem

insgesamt wenigstens N+2 Alternationen über das Durchlassband und Sperrband aufweist. Aufgrund der Zulänglichkeit für eine eindeutige Optimalität ist es möglich, zum Finden eines Filters fortzufahren, der tatsächlich N+2 Alternationen in $E_W^{\text{'}}\left(\omega \right)$

erfüllt. Während dies erfolgt, kann die Tatsache, dass die Frequenzen, bei denen Alternationen in $E_W^{\text{'}}\left(\omega \right)$

auftreten, auch die Alternationsfrequenzen für den angepassten gewichteten Fehler $E_P^{\text{'}}\left(\omega \right)$

für die Frequenzantwort P(e^jω) der Autokorrelationsfunktion sind (d. h. die Fourier-Transformation von p[n]), kann verwendet werden, wobei der Fehler $E_P^{\text{'}}\left(\omega \right)$

ähnlich zu $E_W^{\text{'}}\left(\omega \right)$

definiert ist). Insbesondere wird, falls |H(e^jω)| seinen Extremwert bei einer speziellen Frequenz erzielt und daher eine Alternation in $E_W^{\text{'}}\left(\omega \right)$

bildet, dann die Fourier-Transformation der Autokorrelationsfunktion P(e^jω) = |H(e^jω)|² auch ihren Extremwert erzielen und eine Alternation in dem Fehler $E_P^{\text{'}}\left(\omega \right)$

bei der gleichen Frequenz bilden. Aufgrund der speziellen Wahlen der gewichteten Fehler $E_W^{\text{'}}\left(\omega \right)\text{ und }{E}_P^{\text{'}}\left(\omega \right)$

für |H(e^jω)| bzw. P(e^jω) haben sie eine gleiche Anzahl an Alternationen.The observations and methods described herein will refer to N + 2 alternations, assuming that h [n] is limited to having real-valued coefficients for the sake of simplicity, while the same arguments apply to complex-valued coefficients with 2N + 2 alternations apply. The characterization theorem formulated above requires the adjusted weighted error $e_W^{\text{'}}\left(\omega \right)$

a total of at least N + 2 alternations over the passband and stopband has. Due to the sufficiency of unambiguous optimality, it is possible to proceed to finding a filter that actually has N + 2 alternations in it $e_W^{\text{'}}\left(\omega \right)$

Fulfills. While this is done, the fact that the frequencies at which alternations in $e_W^{\text{'}}\left(\omega \right)$

also occur the alternation frequencies for the adjusted weighted error $e_P^{\text{'}}\left(\omega \right)$

for the frequency response P (e ^jω ) of the autocorrelation function (ie, the Fourier transform of p [n]) can be used, with the error $e_P^{\text{'}}\left(\omega \right)$

similar to $e_W^{\text{'}}\left(\omega \right)$

is defined). In particular, if | H (e ^jω ) | reaches its extreme value at a specific frequency and therefore an alternation in $e_W^{\text{'}}\left(\omega \right)$

then the Fourier transform of the autocorrelation function P (e ^jω ) = | H (e ^jω ) | ² also achieve their extreme value and an alternation in the error $e_P^{\text{'}}\left(\omega \right)$

form at the same frequency. Due to the special choices of weighted mistakes $e_W^{\text{'}}\left(\omega \right)\text{and}{e}_P^{\text{'}}\left(\omega \right)$

for | H (e ^jω ) | or P (e ^jω ) they have an equal number of alternations.

und der Fehler $E_P^{\text{'}}\left(\omega \right)$

die gleiche Anzahl an Alternationen haben, ermöglicht vorteilhafterweise das Ausführen der Gestaltung in der Autokorrelationsdomäne, wobei eine Autokorrelationsfunktion erhalten wird, die die benötigte Anzahl an Alternationen erfüllt, und die Filterkoeffizienten wiedererlangt werden, die diese Funktion als ihre Autokorrelationsfunktion akzeptieren. Das Gestalten einer Autokorrelationsfunktion, die ausreichend Alternationen aufweisen wird, ist viel einfacher als das Gestalten des ursprünglichen Filters, weil die Autokorrelationsfunktion eine Folge mit Phase null ist. Insbesondere ist die Autokorrelation p[n] eines Filters h[n] N-ter Ordnung von der Länge 2N + 1; gerade symmetrisch, falls h[n] realwertige Koeffizienten aufweist, oder konjugiert symmetrisch, falls h[n] komplexwertige Koeffizienten aufweist. Dies ermöglicht, dass ihre Fourier-Transformation P(e^jω) als eine realwertige Funktion ausgedrückt wird, die eine Linearkombination von Kosinus ist, falls gerade symmetrisch, oder eine Linearkombination von Sinus und Kosinus ist, falls konjugiert symmetrisch. In beiden Fällen können das Alternationstheorem und der Remez-Austauschalgorithmus sowie viele in der Technik bekannte andere effiziente Algorithmen verwendet werden, um die optimale Autokorrelationsfolge erfolgreich zu charakterisieren und zu gestalten.Setting up that $e_W^{\text{'}}\left(\omega \right)$

and the mistake $e_P^{\text{'}}\left(\omega \right)$

having the same number of alternations advantageously allows the design to be performed in the autocorrelation domain, obtaining an autocorrelation function satisfying the required number of alternations, and recovering the filter coefficients that accept this function as their autocorrelation function. Designing an autocorrelation function that will have sufficient alternations is much simpler than designing the original filter because the autocorrelation function is a phase zero sequence. In particular, the autocorrelation p [n] of an Nth-order filter h [n] of length 2N + 1; even symmetric if h [n] has real valued coefficients, or conjugate symmetric if h [n] has complex valued coefficients. This allows its Fourier transform P (e ^jω ) to be expressed as a real-valued function, which is a linear combination of cosine, if just symmetric, or a linear combination of sine and cosine, if conjugate symmetric. In both cases, the alternation theorem and the Remez replacement algorithm as well as many other efficient algorithms known in the art may be used to successfully characterize and design the optimal autocorrelation sequence.

Two-step design algorithm

Designing a phase zero sequence that approximates an ideal filter response and treats it as autocorrelation of an FIR filter has been used in the past as a design method for a nonlinear phase FIR filter. However, because design constraints, such as relative weighting on the stopband versus passband, in the autocorrelation domain, due to the squaring relationship between P (e ^jω ) and | H (e ^jω ) | are not the same, the resulting filter does not necessarily reflect the desired weighting. Furthermore, there are none Optimal arguments are available for the final design because the optimality of the autocorrelation sequence for a set of metrics does not make the corresponding filter optimal for the same metrics.

symmetrisch um Eins in dem Durchlassband herum schwingt, d. h. seine Extremwerte zu 1+δ_P und 1-δ_P für einige positive δ_P werden, und (iii) der Maximalwert δ_S von $\left|H\left(e^{i\omega }\right)\right|=\sqrt{P\left(e^{i\omega }\right)}$

in dem Sperrband die gewünschte Gewichtungsbegrenzung, d.h. $\frac{{\delta }_P}{{\delta }_S}=K_{des}$

erfüllt.A characterization algorithm proposed here can be considered to be the proof that one must design the autocorrelation sequence only in such a way that (i) E ' _P (ω) has at least N + 2 alternations, (ii) $\left|H\left(e^{i\omega }\right)\right|=\sqrt{P\left(e^{i\omega }\right)}$

that is, its extreme values become 1 + δ _P and 1-δ _P for some positive δ _P , and (iii) the maximum value δ _S of $\left|H\left(e^{i\omega }\right)\right|=\sqrt{P\left(e^{i\omega }\right)}$

in the stopband the desired weighting limit, ie $\frac{{\delta }_P}{{\delta }_S}=K_{des}$

Fulfills.

die optimale Lösung sein. Die tatsächlichen Filterkoeffizienten können durch spektrales Faktorisieren von p[n] oder eine beliebige andere Technik, die in der Technik bekannt ist, zum Wiedererlangen der ursprünglichen Funktion von ihrer Autokorrelationsfolge erhalten werden, die alle innerhalb des Schutzumfangs der vorliegenden Offenbarung liegen. Es wird mehr als eine Auswahl für das Filter geben, die alle die gleiche Betragsantwort aufweisen, einschließlich einer Minimalphasenauswahl und einer Maximalphasenauswahl unter allen anderen. Die Beziehung zwischen den Extremwerten der Betragsantwort |H(e^jω)| und jenen von P(e^jω)= |H(e^jω)|² sind in 1A und 1B gegeben.The uniqueness of the global minimax optimized magnitude response from the characterization theorem provided above, when such an autocorrelation function is found, then becomes $\left|H\left(e^{j\omega }\right)\right|=\sqrt{P\left(e^{j\omega }\right)}$

be the optimal solution. The actual filter coefficients can be obtained by spectrally factoring p [n] or any other technique known in the art to recover the original function from its autocorrelation sequence, all of which are within the scope of the present disclosure. There will be more than one selection for the filter, all of which have the same magnitude response, including a minimum phase selection and a maximum phase selection among all others. The relationship between the extreme values of the magnitude response | H (e ^jω ) | and those of P (e ^jω ) = | H (e ^jω ) | ² are in 1A and 1B given.

Two steps of the design algorithm can be summarized as follows:

In the first step, an autocorrelation function p [n] is designed such that the Fourier transform of the autocorrelation function, P (e ^jω ), satisfies the properties listed above. Such autocorrelation function may be designed using any suitable method as known in the art. In some embodiments, such an autocorrelation function may be designed using the approach suggested in Section C below.

erfüllt. Beliebige geeignete Verfahren, wie in der Technik bekannt, können verwendet werden, um ein solches Filter zu erhalten, einschließlich unter anderem eines Verfahrens des spektralen Faktorisierens von p[n].In the second step, a filter h [n] is determined such that its magnitude response $\left|H\left(e^{j\omega }\right)\right|\left|H\left(e^{j\omega }\right)\right|=\sqrt{P\left(e^{j\omega }\right)}$

Fulfills. Any suitable methods, as known in the art, may be used to obtain such a filter, including, but not limited to, a method of spectral factorization of p [n].

An algorithm for designing an optimal autocorrelation

in dem Durchlassband und ${\delta }_S^2/2$

in dem Sperrband ist. Da die Autokorrelationsfolge von der Länge 2N+1 und symmetrisch ist, werden wenigstens N+2 Alternationen benötigt, was auch die erforderliche Bedingung für die Optimalität der finalen Gestaltung durch das oben bereitgestellte Charakterisierungstheorem ist. Jedoch sind δ_P and δ_S nicht im Voraus bekannt, so dass die Zielfunktion nicht a priori für den Remez-Austauschalgorithmus bekannt ist.The optimal autocorrelation function or, equivalently, its Fourier transform, such as the in 1B hypothetically using z. For example, the Remez replacement algorithm may be designed to approximate an objective function that $1+{\delta }_{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">P</figure-callout>}}^2$

in the pass band and ${\mathrm{<figure-callout\; id="2"\; label="\delta \; S"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">\delta \; </figure-callout>}}_S^{}/2$

is in the stop band. Since the autocorrelation sequence is of length 2N + 1 and symmetric, at least N + 2 alternations are needed, which is also the required condition for the optimality of the final design by the characterization theorem provided above. However, δ _P and δ _{S are} not known in advance, so the objective function is not known a priori for the Remez interchange algorithm.

One approach that can be used to obtain such an autocorrelation is to choose the objective function to be the ideal filter response, which is one in the passband and zero in the stopband, and to choose the weight on the stopband so that the frequency response of the obtained filter G (e ^jω ) can be scaled and shifted in order, like the Fourier transform, that in 1B to look autocorrelation shown. For this purpose, first a filter g [n] is designed with phase zero, for example using the Parks-McClellan design method, to obtain a frequency response similar to that in FIG 2 shown to be obtained, the Durchlassbandwelligkeitsgröße Δ _P and the Δ _S is Sperrbandwelligkeitsgröße. A scaling coefficient a and a shift coefficient b (hereinafter sometimes collectively referred to as "scaling and shift coefficients") may be selected such that the midpoints of the passband and stopband ranges correspond to those of the autocorrelation in FIG 1B match. Furthermore, the scaling and shift coefficients a and b can be chosen such that the extreme values of the design in FIG 2 with the extreme values of autocorrelation in 1B match after scaling and moving. It is also necessary that the resulting δ _P and δ _S meet the predetermined original weight limit, namely, that: $$\frac{{\delta }_P}{{\delta }_S}=K_{des},$$

With these constraints, a relationship between the weighting K, the at Design the filter in 2 must be used and the resulting ΔP identified as follows: $${\mathrm{\Delta }}_P=\frac{\mathrm{8th}{K}_{des}^2\mathrm{<figure-callout\; id="2"\; label="K\; K"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">K\; </figure-callout>}}{K^{}}$$

The mathematical details about deriving the relationship of equation (10) are provided below.

Equation (10) provides an implicit and non-linear equation that can be solved efficiently, for example, using an iterative procedure. In various embodiments of using an iterative procedure for solving equation (10), the search space on K may be reduced to at each iteration on z. A binary search method or using a Newton-Raphson method. Once the appropriate value is found by K (ie a value of K, for the Δ _P, calculated from G (e ^jω) is equal to the right side of equation (10)), the filter can be obtained by moving and scaling G (e ^jω ) for obtaining the frequency response of the autocorrelation P (e ^jω ) in 1B and then recovering the filter coefficients h [n] as in the second step of the two-step design algorithm described in the previous section.

Overall algorithm for designing a filter

3 FIG. 3 illustrates a flowchart of a method 300 for calculating coefficients h [n] of an FIR filter of order N (hence the filter has N + 1 coefficients) with a minimax optimal magnitude response | H (e ^jω ) | according to some embodiments of the disclosure. The procedure off 3 may be by any suitable processing device, such as, but not limited to, the filter configuration unit 404 , possibly with the help of the processor described below 406 to be executed. Although descriptions of the steps of the procedure 300 which are provided below on the in 4 Referring to the elements shown, is any processing device, the steps of the method 300 in any order, within the scope of the present disclosure.

Before the start of the procedure 300 can the filter configuration unit 404 are supplied with a set of filter configuration parameters as shown in the system illustration 4 as a configuration parameter 412 are shown. Such configuration parameters may specify details of a filter to be designed (ie, to specify a filter for which coefficients h [n] are to be determined as described herein). The configuration parameters may include a filter of order N, the desired passband and stopband frequencies (Ω _P and Ω _S ), a desired magnitude response D (ω) and a desired weighting function W _of (ω). In addition, the configuration parameters may optionally include a default final phase property for the design. For example, although the present disclosure does not limit the phase during calculation of the minimax optimal magnitude response, once such magnitude response is calculated, the designer may choose to obtain the coefficients for the minimum phase filter or the maximum phase filter that indicates the calculated optimum magnitude response. The designer may choose to provide this preference for the final phase property as a configuration parameter. If this input is not provided in advance, the procedure may 300 be adapted to return any or all of the filters that show the calculated optimal magnitude response, and the designer can make a choice at that time, e.g. Via an appropriately designed user interface.

The procedure 300 can start with that the filter configuration unit 404 at 302 K (ie added with Ω _{S, P} and K Ω _of an initial estimate for K set) is initialized. In some embodiments, K may advantageously be chosen such that K ≥ 4K of _the (K _of + 1) holds for a physically meaningful design.

mit der Gewichtungsfunktion $$\text{W}\left(\mathrm{\omega }\right)=\{\begin{array}{cc}\mathrm{1,}& \mathrm{\omega }\in {\mathrm{\Omega }}_p\\ K,& \mathrm{\omega }\in {\mathrm{\Omega }}_s\end{array}.$$

at 304 calculates the filter configuration unit 404 Coefficients of the minimax-optimal even symmetric or conjugate symmetric filter g [n] (with a phase of zero in both cases) of order 2N , The coefficients are calculated to approximate the objective function: $$\text{D}\left(\mathrm{\omega }\right)=\{\begin{array}{cc}\mathrm{1,}& \mathrm{\omega }\in {\mathrm{\Omega }}_p\\ 0& \mathrm{\omega }\in {\mathrm{\Omega }}_s\end{array}$$

with the weighting function $$\text{W}\left(\mathrm{\omega }\right)=\{\begin{array}{cc}\mathrm{1,}& \mathrm{\omega }\in {\mathrm{\Omega }}_p\\ K.& \mathrm{\omega }\in {\mathrm{\Omega }}_s\end{array},$$

For this purpose, the filter configuration unit 404 be configured to use the Remez replacement algorithm, the Parks-McClellan algorithm or any other method as known in the art.

In step 306 can the filter configuration unit 404 G (e ^jω ), calculate the frequency response of g [n] and the maximum value of the passband error Δ _P for this g [n], the latter being equivalent to the maximum value of the absolute weighted error | W (ω) (G (e ^jω ) -D (ω)). is.

In step 308 determines the filter configuration unit 404 whether the value of in step 306 calculated passband error Δ _P the equality in equation ( 10 ) (ie, if the value of in step 306 calculated passband error is equal to the value of the right side of equation (10)). If so, the procedure continues 300 then to the step 312 continued. Otherwise, the filter configuration unit increases 404 , as in 3 with step 310 shown if in step 306 calculated passband error Δ _{P is} less than the value of the right side expression of equation (10), then the value of K, and the method goes to the step 304 back, while, if Δ _{P is} greater than the expression in equation ( 10 ), then the value of K is decreased and the method 300 to the step 304 declining. In various embodiments, the amount to the K in step 310 is increased or decreased, or the boundaries of the search space may be decided in various ways, including methods such as a binary search, the Newton-Raphson method, or any other appropriate method. Regardless of the method, in some embodiments values of K may be chosen such that K ≥ 4K of _the (K _of + 1) holds for a physically meaningful design.

In step 312 can the filter configuration unit 404 calculate the scaling and shift coefficients a and b, e.g. Using equations (24) and (25).

In step 314 can the filter configuration unit 404 the function p [n] from that in step 304 specific function g [n] using the in step 312 calculate calculated scaling and displacement coefficients. In some embodiments, the function p [n] may be calculated as p [n] = a * g [n] + b * δ [n] where δ [n] is the unit pulse function (not to be confused with the passband or stopband ripples of | H (e ^jω ) |, namely δ _P or δ _S ). This p [n] is the autocorrelation sequence that was searched.

In step 316 can the filter configuration unit 404 calculate the coefficients of h [n] based on the autocorrelation sequence p [n] using any method, e.g. Including, inter alia, the spectral factorization method or using the discrete Hilbert transform relationship between the magnitude and phase of a minimum-phase filter, also known as the Bayard-Bode relationship.

There will be more than one filter for which the autocorrelation sequence is p [n]. Depending on which type of phase is dictated by the design (as in the filter configuration unit 404 provided or pre-stored), z. A minimum phase or maximum phase, may be the filter configuration unit 404 be designed to appropriately choose one or more filters, especially if spectral factorization is used. Otherwise, if the Bayard-Bode relationship is used, the resulting filter will be a minimum-phase filter. However, as one skilled in the art will appreciate, other phase characteristics may be obtained from a minimum-phase filter by replacing "zeros" of the filter with their conjugate inverses as desired. Once the positions of their zeroes are determined, the coefficients of filters having different phase characteristics can be calculated using the relationship between a polynomial and its roots, as known in the art.

Derivation of equation ( 10 )

This section presents the derivation of equation (10) provided above.

und $$a\cdot 0+b=\frac{{\delta }_S^2}{2}$$

was Folgendes ergibt: $$a=1+{\delta }_P^2-\frac{{\delta }_S^2}{2}$$

und $$b=\frac{{\delta }_S^2}{2}$$

The scaling coefficient a and the shift coefficient b can be chosen such that the midpoints of the passband and stopband ranges coincide with those of the autocorrelation in FIG 1B match. In particular, applies $$a\cdot 1+b=1+{\delta }_{\mathrm{<figure-callout\; id="2"\; label="P"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">P</figure-callout>}}^2$$

and $$a\cdot 0+b=\frac{{\delta }_{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">S</figure-callout>}}^2}{2}$$

which gives the following: $$a=1+{\delta }_P^2-\frac{{\mathrm{<figure-callout\; id="2"\; label="\delta \; S"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">\delta \; </figure-callout>}}_S^{}}{2}$$

and $$b=\frac{{\delta }_{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">S</figure-callout>}}^2}{2}$$

The relative weight between passband and stopband does not change after scaling the filter response to match that of the autocorrelation, therefore the weights in both can be identical: $$\frac{{\left(1+{\delta }_P\right)}^2-{\left(1-{\delta }^P\right)}^2}{{\mathrm{<figure-callout\; id="2"\; label="\delta \; S"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">\delta \; </figure-callout>}}_S^{}}=\frac{4{\delta }_{\mathrm{<figure-callout\; id="2"\; label="P\; \delta \; S"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">P\; </figure-callout>}}}{{\delta }_S^{}}=\frac{{\mathrm{\Delta }}_P}{{\mathrm{\Delta }}_S}=K$$

gilt, kann K geschrieben werden als $$K=\frac{4K_{des}}{{\delta }_S}$$

oder $${\delta }_S=\frac{4K_{des}}{K}$$

There $\raisebox{1ex}{${\delta }_P$}\!\left/ \!\raisebox{-1ex}{${\delta }_S$}\right.=K_{des}$

is valid, K can be written as $$K=\frac{4K_{des}}{{\delta }_S}$$

or $${\delta }_S=\frac{4K_{des}}{K}$$

To match the upper bound of the filter response in the stopband with that of the autocorrelation after scaling and shifting, the following applies: $${\delta }_S^2=a\cdot {\mathrm{\Delta }}_S+b,$$

aus Gleichung (19) in die Gleichung (20) führt zu Folgendem: $$\frac{16K_{des}^2}{K^2}=\left(1+\frac{16K_{des}^4}{K^2}-\frac{8K_{des}^2}{K^2}\right){\mathrm{\Delta }}_S+\frac{8K_{des}^2}{K^2}$$

Substituting the values of scaling and shift coefficients a and b from equations (15) and (16) and substituting ${\delta }_{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">S</figure-callout>}}^2=\frac{16K_{des}^2}{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">K</figure-callout>}}^2}$

from equation ( 19 ) into the equation ( 20 ) leads to the following: $$\frac{16K_{des}^2}{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">K</figure-callout>}}^2}=\left(1+\frac{16K_{des}^4}{K^2}-\frac{\mathrm{8th}{K}_{des}^2}{K^2}\right){\mathrm{\Delta }}_S+\frac{\mathrm{8th}{K}_{des}^2}{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">K</figure-callout>}}^2}$$

und, da Δ_P = KΔ_S gilt, wird das Folgende erhalten: $${\mathrm{\Delta }}_P=\frac{8K_{des}^{2}K}{K^2+16K_{des}^4-8K_{des}^2}$$

Solving equation ( 21 ) to Δ _S $${\mathrm{\Delta }}_S=\frac{\mathrm{8th}{K}_{des}^2}{K^2+16K_{des}^4-\mathrm{8th}{K}_{des}^2},$$

and, since Δ _P = KΔ _S , the following is obtained: $${\mathrm{\Delta }}_P=\frac{\mathrm{8th}{K}_{des}^2\mathrm{<figure-callout\; id="2"\; label="K\; K"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">K\; </figure-callout>}}{K^{}}$$

Finally, once the appropriate weighting K has been found satisfying this equation, the scaling and shift coefficients can be calculated directly from the parameters of this filter. Using equation (19) and (16), the shift coefficient b can be determined as: $$b=\frac{\mathrm{8th}{K}_{des}^2}{{\text{<figure-callout id="2" label="K" filenames="DE112017002262T5\_0059.png" state="{{state}}">K</figure-callout>}}^2}$$

From equations (24) and (20), the scaling coefficient a can be determined as: $$a=\frac{\mathrm{8th}{K}_{des}^2}{{\text{K}}^2{\mathrm{\Delta }}_S}=\frac{\mathrm{8th}{K}_{des}^2}{\text{K}{\mathrm{\Delta }}_P},$$

System view of improved receiver and digital filter

4 FIG. 12 is a schematic illustration of an exemplary receiver. FIG 400 according to some embodiments of the disclosure. As shown, the receiver 400 a digital filter 402 and a filter configuration unit 404 exhibit. The digital filter 402 is designed to be an input signal 410 to filter that by a receiver 400 receiving it as an input to the digital filter 402 is received to a in 4 shown filtered signal as an output 414 to create. The input signal 410 may be a preprocessed signal, e.g. B. is converted from an analog to a digital domain and is possibly processed to reduce or eliminate noise. The digital filter 402 is designed as a filter with coefficients h [n], these coefficients passing through the filter configuration unit 404 based on configuration parameters 412 calculated for the filter configuration unit 404 to be provided. In various embodiments, at least some of the configuration parameters 412 be provided externally, for. Provided by a user using an appropriate interface, while some of the parameters 412 in the filter configuration unit 404 are pre-stored.

Like also in 4 shown, the receiver can 400 at least one processor 406 and at least one memory element 408 include, along with any other suitable hardware and / or software, its intended functionality of supporting a configuration of the digital filter 402 with appropriate filter coefficients and filtering of the input signal 410 as described here. In some embodiments, the processor 406 Execute software or an algorithm to perform the activities as discussed in this specification, e.g. For example, the processor 406 the algorithm for calculating filter coefficients h [n] by performing steps as described herein, e.g. B. with reference to 3 , To run. Accordingly, the processor can 406 and / or the memory 408 , although they 4 are shown as separate elements as part of the filter configuration unit 404 and / or part of the digital filter 402 to be viewed as.

It should be noted that in order not to overfill the drawing, the receiver 400 Signal processing components of a receiver illustrated and other components that are typically present in receivers, not illustrated. For example, one of ordinary skill in the art would recognize that the receiver 400 and one or more antennas for receiving signals, an integrated circuit capable of providing an analog front end for receiving signals and converting analog input signal into digital data samples of the analog input signal, having various interfaces, ports, etc. In one embodiment, an analog front end may be configured with the processor 406 to communicate to provide digital data samples that the processor 406 for filtering signals with frequency contributions of interest Ω _P , while canceling or reducing subsets to the received signals at frequencies ω ∈ Ω _S other than those in the band of interest or below the noise threshold of the detection mechanism.

The teachings provided herein relate to digital filters configured to filter electromagnetic signals in various frequency ranges (eg, in the radio range, in the optical range, etc.). Furthermore, these teachings relate to the digital filtering of signals received by receivers other than electromagnetic receivers, such as e.g. Sonar receivers.

The processor 406 may be configured to be communicatively coupled to other systems via one or more interconnects or buses. Such a processor may include any combination of hardware, software, or firmware that provides programmable logic, including a microprocessor, a digital signal processor (DSP), a field programmable gate array (FPGA), a Programmable Logic Array (PLA), an Application Specific Integrated Circuit (ASIC), or a Virtual Machine Processor as a non-limiting example. The processor 406 can be communicative with the memory element 408 coupled, for example, in a Direct Memory Access (DMA) configuration. Such a memory element may include any volatile or non-volatile memory technology, including Double Data Rate (DDR) random access memory (RAM), Synchronous RAM (SRAM), Dynamic RAM (DRAM), Flash, Read only memory (ROM ), optical media, virtual storage, magnetic or tape storage, or any other suitable technology. Any of the memory objects discussed herein should be construed as included within the broad term "memory element." The information in the digital filter 402 , the filter configuration unit 404 , the processor 406 or the memory 408 tracked or sent to, could be provided in any database, register, control list, cache or memory structure, all of which can be referenced at any suitable time frame. Any such storage options may be included within the broad term "storage element" as used herein. Likewise, any of the potential processing elements, modules and machines described herein should be construed as included within the broad term "processor". Each of the in 4 shown elements, eg. B. the digital filter 402 and the filter configuration unit 404 , may also have suitable interfaces for receiving, transmitting and / or otherwise communicating data or information in a network environment.

In certain implementation examples, mechanisms for supporting the configuration of the digital filter may be provided 402 and for filtering the input signal 410 as outlined herein, be implemented by a logic encoded in one or more tangible media that may include nonvolatile media, e.g. Embedded logic provided in an ASIC, DSP instructions, software (possibly including object code and source code) to be executed by a processor or other similar machine, and so on. In some of these cases, memory elements such as e.g. B. the in 4 shown memory 408 Store data or information used for these operations described here. This includes memory elements that are capable of storing software, logic, code, or processor instructions that are executed to perform the activities described herein. A processor may execute any type of instructions associated with the data or information to accomplish the operations described in detail herein. In one example, the processors, such as e.g. B. the in 4 shown processor 406 , transform an item or article (eg, data) from one state or object to another state or object. In another example, the activities outlined herein may be implemented with fixed logic or programmable logic (eg, software / computer instructions executed by a processor), and the elements identified herein could be any type of programmable processor, programmable digital logic (eg an FPGA, a DSP, an erasable one programmable read only memory (EPROM), an electrically erasable programmable read only memory (EEPROM), or an ASIC having digital logic, software, code, electronic instructions, or any suitable combination thereof.

Exemplary data processing system

5 FIG. 12 is a block diagram illustrating an exemplary data processing system. FIG 500 in accordance with an embodiment of the present disclosure. Such a data processing system could be embodied in any system designed to implement the FIR design techniques described herein.

As in 5 shown, the data processing system 500 at least one processor 502 have, by a system bus 506 with memory elements 504 is coupled. As such, the data processing system may program code within the memory elements 504 to save. Furthermore, the processor can 502 Execute program code on the memory elements 504 via a system bus 506 is accessed. In one aspect, the data processing system may be implemented as a computer suitable for storing and / or executing program code. It is understood, however, that the data processing system 500 may be implemented in the form of any system including a processor and a memory capable of performing the functions described within this description.

The memory elements 504 may include one or more physical storage devices, such as, for example, a local storage 508 and one or more mass storage devices 510 , The local memory may refer to a random access memory or other non-persistent storage device (s) commonly used during actual execution of the program code. A mass storage device may be implemented as a hard disk or other persistent data storage device. The processing system 500 may include one or more caches (not shown) that may provide temporary storage of at least some program code to reduce the number of times the program code executes out of the mass storage device 510 must be retrieved.

Input / output (I / O) devices acting as an input device 512 and an output device 514 can optionally be coupled to the data processing system. Examples of input devices may include, but are not limited to, a keyboard, a display device such as a mouse, or the like. Examples of output devices may include, but are not limited to, a monitor or display, speakers, or the like. Input and / or output devices may be coupled to the data processing system either directly or through intervening I / O controls.

In one embodiment, the input and output devices may be implemented as a combined input / output device (in FIG 5 illustrated with a dashed line representing the input device 512 and the dispenser 514 surrounds). An example of such a combined device is a touch-sensitive display, sometimes referred to as a _" touch screen display" or simply "touch screen". In such embodiments, an input to the device may be caused by a movement of a physical object, such as e.g. A stylus or a finger of a user, on or near the touch screen display.

A network adapter 516 Optionally, it may also be coupled to the data processing system to enable it to be coupled to other systems, computer systems, remote network devices, and / or remote storage devices through intervening private or public networks. The network adapter may be a data receiver for receiving data through the systems, devices and / or networks to the data processing system 500 and a data transmitter for transmitting data from the data processing system 500 to the systems, devices and / or networks. Modems, cable modems, and Ethernet cards are examples of different types of network adapters that work with the data processing system 500 can be used.

As in 5 illustrates the memory elements 504 an application 518 to save. In various embodiments, the application 518 in the local store 508 the one or more mass storage devices 510 or stored separately from the local memory and the mass storage devices. It is understood that the data processing system 500 furthermore a (in 5 not shown), which is an execution of the application 518 can allow. The application 518 , which is implemented in the form of executable program code, may be processed by the data processing system 500 , z. B. by the processor 502 to be executed. In response to running the application, the Data processing system 500 be configured to perform one or more operations described herein or method steps.

Selected examples

In a first set of examples, Examples A, Example 1A presents a computer-implemented method for determining coefficients h [n], where n = 0, 1, ..., N, of an FIR digital filter (FIR: finite impulse response) Order N having an magnitude response adapted to approximate a response of an ideal FIR filter configured to pass components of signals at frequencies ω within a set of passband frequencies Ω _P and components of signals at frequencies within suppressing a set of stop band frequencies Ω _S such that a ratio of an error in the amount of pass band frequencies Ω _P (ie, an error δ _P for frequencies ω ∈ Ω _P ) and an error in the amount of stop band frequencies Ω _S (ie, an error δ _S for frequencies ω ∈ Ω _S) is equal to K _{of the} or lies in a tolerance range of this, said method comprising: initializing a value e ines scalar K; Determining coefficients g [n] of a minimax-optimal even-symmetric or conjugate-symmetric filter (even symmetric if h [n] is limited to real-valued coefficients and conjugate symmetric if h [n] can take complex-valued coefficients) of order 2N, to approximate a response of a FIR filter configured to pass components of signals at frequencies within the set of passband frequencies Ω _P and to suppress components of signals at frequencies within the set of stopband frequencies Ω _S , such that a ratio of an error in the set of passband frequencies Ω _P (ie an error Δ _P for frequencies ω ∈ Ω _P ) and an error in the set of stopband frequencies Ω _S (ie an error Δ _S for frequencies ω ∈ Ω _S ) equals a tolerance range of K or lies in this; Determining a frequency response G (e ^jω ) of the minimax optimal even symmetric or conjugate symmetric filter with the determined coefficients g [n]; Determining a maximum passband error Δ _P from the determined frequency response G (e ^jω ); Determining whether the determined maximum passband error Δ _P based satisfies a predetermined condition with respect to a comparison value based on values of the scalars K and K _of; and when the determined maximum passband error Δ _P satisfies the condition with respect to the comparison value, calculating a set of values p [n] by scaling the determined coefficients g [n] by a first scaling value a and adding a unit pulse δ [n] to the scaled coefficients a * g [n] and determine the coefficients h [n] as a set of (N + 1) values for which the set of values p [n] is an autocorrelation sequence ,

Example 2A provides the method of Example 1A, wherein determining the coefficients g [n] of the minimax optimal even symmetric or conjugate symmetric filter includes determining the coefficients g [n] using a Remez replacement algorithm or a Parks-McClellan algorithm ,

Example 3A provides the method of Examples 1A or 2A wherein determining the frequency response G (e ^jω ) of the minimax-optimal even-symmetric or conjugate-symmetric filter includes computing a Fourier transform of the determined coefficients g [n].

Example 4A provides the method of any preceding Example A, wherein determining the maximum passband error Δ _P includes determining a maximum absolute deviation of a value of G (e ^jω ) from a value of 1 for frequencies within the amount of passband frequencies Ω _P.

ist.Example 5A provides the method according to any one of the preceding examples A, wherein the comparison value is a value that is indicative (eg, equal or proportional to) $\frac{\mathrm{8th}{\text{K}}_{\text{of}}^2\text{<figure-callout id="2" label="K K" filenames="DE112017002262T5\_0059.png" state="{{state}}">K </figure-callout>}}{{\text{K}}^{}}$

is.

Example 6A provides the method of any of Examples 1A-5A, the condition with respect has to the comparison value, a difference between the determined maximum passband error Δ _P and the comparative value is within a predetermined range with respect to 0.

Example 7A provides the method of any of Examples 1A-5A, the condition with respect has to the comparison value, a ratio between the determined maximum passband error Δ _P and the comparative value is within a predetermined range with respect to 1.

Example 8A provides the method of any one of Examples 1A-5A, wherein the condition with respect to the comparison value includes that the determined maximum passband error Δ _P equals the comparison value.

ist.Example 9A provides the method according to any one of the preceding examples A, wherein the scaling coefficient a equals $\frac{\mathrm{8th}{\text{K}}_{\text{of}}^2}{\text{K}{\mathrm{\Delta }}_{\text{P}}}$

is.

ist. Example 10A provides the method according to any one of the preceding examples A, wherein the shift coefficient b is equal to $\frac{\mathrm{8th}{\text{K}}_{\text{of}}^2}{{\text{<figure-callout id="2" label="K" filenames="DE112017002262T5\_0059.png" state="{{state}}">K</figure-callout>}}^2}$

is.

Example 11A provides the method according to any of the preceding examples A, wherein determining the coefficients h [n] as the set of (N + 1) values for which the set of values p [n] is the autocorrelation sequence, performing a spectral Factoring on the set of values p [n] or using a discrete Hilbert transform relationship between the magnitude response and the phase response of a minimum-phase filter, also known as Bayard-Bode relationship.

Example 12A provides the method of any one of preceding examples A, wherein determining the coefficients h [n] as the set of (N + 1) values for which the set of values p [n] is the autocorrelation sequence, determining two or more sets of the (N + 1) values for which the set of values p [n] is the autocorrelation sequence, wherein each set of the (N + 1) values is associated with a different phase property.

Example 13A provides the method of Example 12A, further comprising selecting a set of (N + 1) values from the two or more sets of the (N + 1) values for which the set of values p [n] represents the autocorrelation sequence a special live feature.

Example 14A provides the method of Example 13A, with the particular phase characteristic having a minimum phase or phase.

Example 15A illustrates the method according to one of the preceding Examples A ready, the method if the determined maximum passband error Δ _P does not satisfy the condition with respect to the reference value, further comprising: performing iterations of changing a value of the scalar K from that for which the determined maximum passband error Δ _P has not satisfied the condition with respect to the comparison value, and repeating, for the changed value of the scalar K, the determination of the coefficients g [n] of the minimax-optimal even symmetric or conjugate symmetric filter, determining the frequency response G (e ^jω ), determining the maximum passband error Δ _P, and determining whether the determined maximum passband error Δ _P satisfies the condition with respect to the comparison value until the determined maximum passband error Δ _P satisfies the condition with respect to Comparison value met.

Methods by which to increase or decrease K include, but are not limited to, binary search methods or the Newton-Raphson method, etc., all of which are within the scope of the present disclosure.

In a second set of examples, Examples B, Example 1B provides a system for filtering a signal received by an electromagnetic receiver. The system includes a filter configuration unit and a digital filter. The filter configuration unit is for configuring a first order N filter to be applied to the signal by determining coefficients h [n] of the first filter, where a target response of the first filter is represented by a target (desired) response function D (ω) and a variable K _des indicative of a target ratio of an error in an amount of pass band frequencies Ω _P and an error in an amount of stop band frequencies Ω _{S of} the first filter is given. The filter configuration unit is configured to determine the coefficients of the first filter by repeating steps i) determining a variable K indicative of a ratio of an error in the amount of passband frequencies Ω _P and an error in the amount of stopband frequencies Ω _{S of} a second one is filter of order 2N which (by the target response function D ω) and the variable K is set to a new value, and ii) determining coefficients of g [n] of the second filter having a target response, until a maximum passband error Δ _P a frequency response G (e ^jω ) of the second filter satisfies a condition with respect to a comparison value based on the variables K and K _of FIG. The filter configuration unit is further configured to determine the coefficients h [n] of the first filter based on the coefficients g [n] of the second filter. The digital filter is configured to then generate a filtered signal by applying the first filter having the calculated coefficients h [n] to the signal.

Example 2B provides the system of Example 1A, where the second filter is a minimax-optimal even-symmetric filter and the coefficients h [n] of the first filter have real values.

Example 3B provides the system of Example 1A wherein the second filter is a minimax optimal conjugate symmetric filter and the coefficients h [n] of the first filter have complex values.

Example 4B provides the system according to any of the preceding examples B, wherein determining the coefficients h [n] of the first filter based on the coefficients g [n] of the second filter calculating a set of values p [n] by scaling the coefficients g [n] of the second filter having a scaling coefficient a and adding a unit pulse δ [n] scaled with a shift coefficient b, and determining the coefficients h [n] of the first filter as a set of (N + 1) values for which the calculated set of values p [ n] is an autocorrelation sequence.

ist.Example 5B provides the system of Example 4B, where the scaling coefficient a equals $\frac{\mathrm{8th}{K}_{des}^2}{K{\Delta }_P}$

is.

ist.Example 6B provides the system according to Examples 4B or 5B, where the shift coefficient b is equal to $\frac{\mathrm{8th}{K}_{des}^2}{{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">K</figure-callout>}}^2}$

is.

einschließt.Example 7B provides the system according to one of Examples 4B-6B, wherein determining the coefficients h [n] of the first filter is the set of (N + 1) values for which the set of values p [n] is the autocorrelation sequence Performing a spectral factorization on the set of values p [n] or applying a Bayard-Bode relation to a square root of a frequency response of the set of values p [n], namely $\left|H\left(e^{i\omega }\right)\right|=\sqrt{P\left(e^{i\omega }\right)}.$

includes.

Example 8B provides the system according to one of Examples 4B-7B, wherein determining the coefficients h [n] as the set of (N + 1) values for which the set of values p [n] is the autocorrelation sequence determines two or more sets of the (N + 1) values for which the set of values p [n] is the autocorrelation sequence, where each set of the (N + 1) values is associated with a different phase property.

Example 9B provides the system of Example 6B, further comprising selecting a set of (N + 1) values from the two or more sets of the (N + 1) values for which the set of values p [n] represents the autocorrelation sequence a special live feature.

Example 10B provides the system of Example 9B, with the particular phase characteristic having a minimum phase or phase.

Example 11B provides the system of any preceding Example B, wherein determining the coefficients g [n] of the second filter includes using a Remez replacement algorithm or a Parks-McClellan algorithm.

Example 12B provides the system of any preceding Example B, wherein determining the maximum passband error Δ _P determines a value indicative of a maximum absolute deviation of the frequency response G (e ^jω ) of the second filter from a value of 1 for frequencies within the amount of passband frequencies Ω _P includes.

ist.Example 13B provides the system according to any of the preceding examples B, wherein the comparison value is a value indicative of $\frac{\mathrm{8th}{K}_{des}^2\mathrm{<figure-callout\; id="2"\; label="K\; K"\; filenames="DE112017002262T5\_0059.png"\; state="\{\{state\}\}">K\; </figure-callout>}}{K^{}}$

is.

Example 14B provides the system of any one of Examples 1B-13B, wherein the condition with respect to the comparison value comprises a difference between the maximum passband error _ΔP and the comparison value that is within a range with respect to zero.

Example 15B provides the system according to any one of Examples 1B-13B, wherein the condition with respect to the comparison value comprises a ratio between the maximum passband error _ΔP and the comparison value that is within a range with reference to FIG.

Example 16B provides the system according to one of Examples 1B-13B, the condition with respect to the reference value having the maximum passband error Δ _P, which is equal to the comparison value.

Example 17B provides the system according to one of the preceding examples B, wherein the target response of the second filter is a response of the second filter for which a ratio of an error due to a difference between the response of the second filter and the target response function D (ω) in the Amount of passband frequencies Ω _P and an error due to a difference between the response of the second filter and the target response function D (ω) in the set of stopband frequencies Ω _S equal to the variable K is within or within a tolerance of this.

In another example B according to one of the preceding examples B, the target response of the first filter is a response of the first filter for which a ratio of an error due to a difference between the response of the first filter and the target response function D (ω) in the set of passband frequencies ω _P and an error due to a difference between the response of the first filter and the target response function D (ω) is in the amount of stop band frequencies ω _S equal to the variable K _of or within a tolerance range of this.

Example 18B provides the system according to any one of the preceding examples B, wherein both the first filter and the second filter is a finite impulse response (FIR) filter.

Example 19B provides a computer-implemented method of operating a digital filter. The method comprises calculating coefficients h [n] of a first filter of order N, wherein a target response of the first filter through a target response function D (ω) and a variable K _des indicative of a target ratio of an error in an amount of passband frequencies Ω _P and an error in a set of stopband frequencies Ω _{S of} the first filter is given by performing one or more iterations of i) determining a variable K indicative of a ratio of an error in the set of passband frequencies Ω _P and one Error in the amount of stop band frequencies Ω _{S of} a second filter of order 2N is to a new value and ii) determining coefficients g [n] of the second filter, where a target response of the second filter is given by the target response function D (ω) and the variable K is given, the iterations are performed until a maximum passband error Δ _{P of} a frequency response G (e ^jω ) of the second filter satisfies a condition with respect to a comparison value based on the variables K and K _of FIG. The method further comprises: determining the coefficients h [n] of the first filter based on the coefficients g [n] of the second filter; and configuring the digital filter to apply the first filter having the coefficients h [n] to a signal to produce a filtered signal.

Example 20B provides the method of Example 19B, further comprising receiving a value of the order N, the target response function D (ω) and the variable K _of via a user interface.

Further examples provide the methods of Examples 19B or 20B, the methods further comprising steps of operating the system of any one of Examples 1B-18B.

Other examples provide a system having means for implementing the method according to any one of the preceding examples, a computer program adapted to implement the method according to any one of the preceding examples, one or more non-volatile tangible media encoding a logic containing instructions for Embodiments which, when executed by a processor, are operable to perform operations of the method according to any one of the preceding examples, and a system comprising at least one memory element adapted to store computer-executable instructions and at least one processor that is coupled to the at least one memory element and configured to execute the instructions to perform the method according to any one of the preceding examples.

Variations and implementations

It is noted that throughout this description, references to various features (eg, elements, structures, modules, components, steps, acts, characteristics, etc.) described in "one embodiment," "an embodiment," "one embodiment." , "Another embodiment", "some embodiments", "various embodiments", "other embodiments", "an alternative embodiment" and the like are intended to mean that any such features are included in one or more embodiments of the present disclosure, possibly combined in the same embodiments or not necessarily combined.

In one embodiment, portions or entire electrical circuits of the figures may be implemented on a motherboard of an associated electronic device. The motherboard may be a general printed circuit board having various components of the internal electronic system of the electronic device and may further provide connectors for other peripheral devices. More specifically, the motherboard can provide the electrical connections through which other components of the system can communicate electrically. Any suitable processors (including digital signal processors, microprocessors, sub-chip sets, etc.), memory elements, etc. may be suitably coupled to the motherboard based on particular configuration requirements, processing requirements, computer design, and so on. Other components, such as external memory, additional sensors, audio / video display controls, and peripheral devices may be attached to the motherboard as plug-in cards, via cables, or integrated into the motherboard itself.

In another embodiment, portions or all of the electrical circuits of the figures may be implemented as stand-alone modules (eg, a device with associated components and circuitry configured to perform a particular application or function) or as a plug-in Modules can be implemented in application-specific hardware of electronic devices. It is noted that certain embodiments of the present disclosure may be readily incorporated into a system on chip (SOC) package, either partially or entirely. An SOC represents an IC that integrates components of a computer or other electronic system into a single chip. It can have digital, analog, mixed signal and often high frequency functions: all of them can be provided on a single chip substrate. Other embodiments may include a multi-chip module (MCM) having a plurality of separate ICs located within a single electronic housing and configured to interact with each other through the electronic housing. In various other embodiments, the gain functionalities in one or more silicon cores may be implemented in application specific integrated circuits (ASICs), field programmable gate arrays (FPGAs), and other semiconductor chips.

It is also imperative to note that all of the specifications, dimensions, and relationships outlined herein (eg, the number of processors and memory elements, logic operations, etc.) are provided for exemplary and instructional purposes only. Such information may be varied considerably without departing from the spirit of the present disclosure or the scope of the appended claims. The specifications are for a non-limiting example only and should be construed as such. In the foregoing description, exemplary embodiments have been described with reference to certain processor and / or component arrangements. Various modifications and changes may be made to such embodiments without departing from the scope of the appended claims. The description and drawings are accordingly to be construed in an illustrative rather than a limiting sense.

It should be noted that with the numerous examples provided herein, interactions may be described in terms of two, three, four or more electrical components. However, this was done for purposes of clarity and example only. It will be understood that the system may be composed in any suitable manner. Along with similar design elements, any of the illustrated components, modules, and elements of the figures may be combined in various possible configurations, all of which are clearly within the broad scope of this description. In certain cases, it may be easier to describe one or more of the functionalities of a given set of flows merely by reference to a limited number of electrical elements. It will be understood that portions of or all of the electrical circuits of the figures and their teachings are readily scalable and capable of accommodating a large number of components as well as more complicated / complex arrangements and configurations. Accordingly, the examples provided should not limit the scope of protection or inhibit the extensive teachings of the parts or all of the electrical circuitry that may be applied to a variety of other architectures.

Numerous other changes, substitutions, variations, changes, and modifications may be determined by one of ordinary skill in the art, and it is intended that the present disclosure encompass all such changes, substitutions, variations, changes, and modifications as fall within the scope of the appended claims. Although the claims may be presented in a single dependency format in the style used prior to the USPTO, it should be understood that any claim may depend on or be combined with any preceding claim of the same type unless clearly technically unworkable ,

Claims (25)

A system for filtering a signal received by an electromagnetic receiver, the system comprising: a filter configuration unit for configuring a first order N filter to be applied to the signal by determining coefficients h [n] of the first filter, wherein a target response of the first filter is by a target response function D (ω) and a variable K _des indicative of a target ratio of an error in an amount of passband frequencies Ω _P and an error in an amount of stopband frequencies Ω _{S of} the first filter, is given; and a digital filter for generating a filtered signal by applying the first filter to the signal, the filter configuration unit configured to determine the coefficients h [n] of the first filter based on coefficients g [n] of a second filter of order 2N. System after Claim 1 , where the second filter is a minimax optimal even symmetric filter. System after Claim 1 , where the second filter is a minimax-optimal conjugate symmetric filter. System according to one of Claims 1 - 3 wherein determining the coefficients h [n] of the first filter based on the coefficients g [n] of the second filter comprises: calculating a set of values p [n] by scaling the coefficients g [n] of the second filter with a scaling coefficient a and adding a unit pulse δ [n] scaled with a shift coefficient b, and determining the coefficients h [n] of the first filter as a set of (N + 1) values for which the set of values p [n] is an autocorrelation sequence is. System after Claim 4 wherein determining the coefficients h [n] of the first filter as the set of (N + 1) values includes performing spectral factorization on the set of values p [n]. System after Claim 4 wherein determining the coefficients h [n] as the set of (N + 1) values includes determining two or more sets of the (N + 1) values for which the set of values p [n] is the autocorrelation sequence, each set of (N + 1) values is associated with a different phase property. System after Claim 6 further comprising selecting a set of (N + 1) values from the two or more sets of the (N + 1) values for which the set of values p [n] is the autocorrelation sequence having a specific phase characteristic. System after Claim 7 , wherein the specific phase characteristic has a minimum phase or a maximum phase. System according to one of Claims 1 - 3 where the coefficients g [n] of the second filter are determined using a Remez interchange algorithm or a Parks-McClellan algorithm. System according to one of Claims 1 - 3 wherein the filter configuration unit is further configured to determine the coefficients of the first filter based on coefficients of a second filter of order 2N by: repeating steps i) establishing a variable K indicative of a ratio of an error in the amount of passband frequencies Ω _P and an error in the amount of stop band frequencies Ω _{S of} the second filter is to a new value and ii) determining coefficients g [n] of the second filter with a target response given by the target response function D (ω) and the variable K until a maximum passband error Δ _P a frequency response g ^(jω e) of the second filter is a condition with respect to a comparison value to the variable K and K _of met based, and determining the coefficients h [n] of the first filter based on the coefficients g [n] of the second filter. System after Claim 10 wherein determining the coefficients h [n] of the first filter based on the coefficients g [n] of the second filter comprises: calculating a set of values p [n] by scaling the coefficients g [n] of the second filter with a scaling coefficient a and adding a unit pulse δ [n] scaled with a shift coefficient b, and determining the coefficients h [n] of the first filter as a set of (N + 1) values for which the set of values p [n] is an autocorrelation sequence is. System after Claim 11 where the scaling coefficient a equals $\frac{\mathrm{8th}{K}_{des}^2}{K{\Delta }_P}$

is. System after Claim 11 , where the shift coefficient b is equal to $\frac{\mathrm{8th}{\text{K}}_{\text{of}}^2}{{\text{K}}^2}$

is. System after Claim 10 wherein determining the maximum passband error Δ _P includes determining a value indicative of a maximum absolute deviation of the frequency response G (e ^jω ) of the second filter from a value of 1 for frequencies within the set of passband frequencies Ω _P. System after Claim 10 , where the comparison value is a value indicative of $\frac{\mathrm{8th}{K}_{des}^{2}K}{K^2+16K_{des}^4-\mathrm{8th}{K}_{des}^2}$

is. System after Claim 10 wherein the condition with respect to the comparison value comprises a difference between the maximum passband error _ΔP and the comparison value that is within a range with respect to zero. System after Claim 10 wherein the condition with respect to the comparison value comprises a ratio between the maximum passband error _ΔP and the comparison value that is within a range with reference to FIG. System after Claim 10 wherein the condition with respect to the comparison value comprises the maximum passband error Δ _P equal to the comparison value. System after Claim 10 wherein the target response of the second filter comprises: a response of the second filter for which a ratio of an error due to a difference between the response of the second filter and the target response function D (ω) in the amount of passband frequencies Ω _P and an error due to a difference between the response of the second filter and the target response function D (ω) in the set of stopband frequencies Ω _{S is} equal to or within a tolerance of the variable K. System according to one of Claims 1 - 3 where both the first filter and the second filter is a finite impulse response (FIR) filter. A computer-implemented method of operating a digital filter, the method comprising: calculating coefficients h [n] of a first filter of order N, wherein a target response of the first filter by a target response function D (ω) and a variable K _des indicative of is a target ratio of an error in an amount of passband frequencies Ω _P and an error in an amount of stopband frequencies Ω _{S of} the first filter, is specified by performing one or more iterations of i) setting a variable K to a new value that is indicative for a ratio of an error in the amount of passband frequencies Ω _P and an error in the amount of stopband frequencies Ω _{S of} a second filter of order 2N, and ii) determining coefficients g [n] of the second filter, wherein a target response of the second filter is given by the target response function D (ω) and the variable K, the iterations d are urchgeführt until a maximum passband error Δ _P a frequency response G (e ^jω) of the second filter based satisfies a condition with respect to a comparison value to the variable K and K _of; Determining the coefficients h [n] of the first filter based on the coefficients g [n] of the second filter; and configuring the digital filter to apply the first filter having the coefficients h [n] to a signal to produce a filtered signal. Method according to Claim 20 further comprising receiving a value of the order N, the target response function D (ω) and the variable K _of via a user interface. Method according to Claims 21 or 22 wherein determining the coefficients h [n] of the first filter based on the coefficients g [n] of the second filter comprises: calculating a set of values p [n] by scaling the coefficients g [n] of the second filter with a scaling coefficient a and adding a unit pulse δ [n] scaled with a shift coefficient b, and determining the coefficients h [n] of the first filter as a set of (N + 1) values for which the set of values p [n] is an autocorrelation sequence is. Method according to Claim 23 where the scaling coefficient a equals $\frac{\mathrm{8th}{K}_{des}^2}{K{\Delta }_P}$

is. Method according to Claim 23 , where the shift coefficient b is equal to $\frac{\mathrm{8th}{K}_{des}^2}{K^2}$

is.

Images